Gauss–Jordan elimination methods for the Moore–Penrose inverse of a matrix

We present an alternative explicit expression for the Moore–Penrose inverse of a matrix. Based on this expression, we propose a Gauss–Jordan elimination method for the computation of A†. Its computational complexity indicates that this method is more efficient than the existing Gauss–Jordan elimination method in the literature for a large class of problems. An example is included to illustrate the new method.